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Let $A$ be an $N \times N$ irreducible matrix with entries in $\{0,1\}$. We define the topological Markov Dyck shift $D_A$ to be a nonsofic subshift consisting of bi-infinite sequences of the $2N$ brackets $(_1,\dots,(_N,)_1,\dots,)_N$ with both standard bracket rule and Markov chain rule coming from $A$. It is regarded as a subshift defined by the canonical generators $S_1^*,\dots, S_N^*, S_1,\dots, S_N$ of the Cuntz-Krieger algebra $\mathcal{O}_A$. We construct an irreducible $\lambda$-graph system $\mathcal{L}^{{\mathrm{Ch}}(D_A)}$ that presents the subshift $D_A$ so that we have an associated simple purely infinite $C^*$-algebra $\mathcal{O}_{\mathcal{L}^{{\mathrm{Ch}}(D_A)}}$. We prove that $\mathcal{O}_{\mathcal{L}^{{\mathrm{Ch}}(D_A)}}$ is a universal unique $C^*$-algebra subject to some operator relations among $2N$ generating partial isometries.

$C^*$-algebras arising from Dyck systems of topological Markov chains